Ellis, Filmus, and Friedgut proved an old conjecture of Simonovits and Sós showing that any triangle-intersecting family of graphs on n vertices has size at most $2^{\left(\begin{array}{c}n\\ 2\end{array}\right)-3}$, with equality for the family of graphs containing some fixed triangle. They conjectured that their results extend to cross-intersecting families, as well to $K_t$-intersecting families. We prove these conjectures for $t\in \{3,4\}$, showing that if ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are families of graphs on n labeled vertices such that for any $G_1\in {\mathcal{F}}_1$ and $G_2\in {\mathcal{F}}_2$, $G_1\cap G_2$ contains a $K_t$, then $|{\mathcal{F}}_1||{\mathcal{F}}_2|\le 4^{\left(\begin{array}{c}n\\ 2\end{array}\right)-\left(\begin{array}{c}t\\ 2\end{array}\right)}$, with equality if and only if ${\mathcal{F}}_1={\mathcal{F}}_2$ consists of all graphs that contain some fixed $K_t$. We also establish a stability result. More generally, “$G_1\cap G_2$ contains a $K_t$” can be replaced by “$G_1$ and $G_2$ agree on a non-$(t-1)$-colorable graph.”

Ellis、Filmus 和 Friedgut 证明了 Simonovits 和 Sós 的一个古老猜想，表明任何在n 个顶点上的三角形相交图族的大小至多$2^{（\begin{array}{c}n\\ 2\end{array}）-3}$，对于包含某个固定三角形的图族来说是相等的。他们推测他们的结果适用于交叉家庭，也适用于$K_t$- 交叉的家庭。我们证明这些猜想$t\epsilon \{3,4\}$，表明如果${\mathcal{F}}_1$和${\mathcal{F}}_2$是n 个标记顶点上的图族，对于任何$G_1\epsilon {\mathcal{F}}_1$和$G_2\epsilon {\mathcal{F}}_2$,$G_1\cap G_2$包含一个$K_t$， 然后$|{\mathcal{F}}_1||{\mathcal{F}}_2|\le 4^{（\begin{array}{c}n\\ 2\end{array}）-（\begin{array}{c}t\\ 2\end{array}）}$，相等当且仅当${\mathcal{F}}_1={\mathcal{F}}_2$由所有包含一些固定的图组成$K_t$。我们还建立了稳定性结果。更普遍， ”$G_1\cap G_2$包含一个$K_t$” 可以替换为 “$G_1$和$G_2$同意非$（t-1）$-可着色的图表。”